The Joint Translation Spectrum and Manhattan Manifolds: Generalizing the Study of Group Actions on Metric Spaces

Core Concepts

This research paper introduces the concept of joint translation spectrum and Manhattan manifolds, providing a novel framework to analyze and compare group actions on metric spaces, particularly focusing on Gromov-hyperbolic groups.

Abstract

• Bibliographic Information: Cantrell, S., Reyes, E., & Sert, C. (2024). THE JOINT TRANSLATION SPECTRUM AND MANHATTAN MANIFOLDS. arXiv preprint arXiv:2411.06375v1.
• Research Objective: This paper aims to introduce and investigate the geometric analogs of the Benoist limit cone and joint spectrum, termed the translation cone and joint translation spectrum, respectively. These concepts provide a powerful toolset for comparing various isometric actions on metric spaces, quasi-morphisms, and other related mathematical objects.
• Methodology: The authors utilize techniques from geometric group theory, ergodic theory, and dynamical systems to establish the properties and characterizations of the joint translation spectrum. They leverage the concept of hyperbolic metric potentials, which serve as a generalization of metrics on Gromov-hyperbolic groups. Additionally, they introduce Manhattan manifolds, higher-dimensional analogs of the Manhattan curve, to analyze the joint translation spectrum.
• Key Findings: The paper demonstrates that the joint translation spectrum is a convex compact set that captures the asymptotic behavior of hyperbolic metric potentials. It establishes a connection between the joint translation spectrum and random walks on Gromov-hyperbolic groups, showing that the closure of the random walk spectrum coincides with the joint translation spectrum. Furthermore, the authors prove that the Manhattan manifold, defined as the graph of a specific critical exponent function, is a C1-manifold and its gradient function provides a homeomorphism to the interior of the joint translation spectrum.
• Main Conclusions: This research provides a novel framework for studying group actions on metric spaces by introducing the joint translation spectrum and Manhattan manifolds. The established connections between these concepts and random walks, dynamical systems, and the geometry of metric structures offer new insights into the asymptotic behavior of group actions.
• Significance: This work significantly contributes to the field of geometric group theory by introducing new tools and perspectives for analyzing group actions. The findings have implications for understanding the geometry of metric spaces, the dynamics of group actions, and the behavior of random walks on groups.
• Limitations and Future Research: The paper primarily focuses on Gromov-hyperbolic groups. Exploring the extension of these concepts to a broader class of groups and their actions could be a potential avenue for future research. Additionally, investigating the applications of the joint translation spectrum and Manhattan manifolds in other areas of mathematics and related fields could yield further insights.

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The joint translation spectrum and Manhattan manifolds

by Stephen Cant... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06375.pdf

Deeper Inquiries

How can the concept of joint translation spectrum be extended to study group actions on non-hyperbolic metric spaces?

Extending the concept of the joint translation spectrum to non-hyperbolic metric spaces presents significant challenges, primarily because many of the tools and properties used in the hyperbolic setting do not directly translate. Here's a breakdown of the challenges and potential approaches:
Challenges:

Lack of Hyperbolic Properties: Hyperbolicity provides crucial properties like the stability of quasi-geodesics and the well-defined nature of the Gromov boundary, which are fundamental to defining the joint translation spectrum. These properties are generally absent in non-hyperbolic spaces.
Behavior of Metric Potentials: The notion of hyperbolic metric potentials heavily relies on the geometry of hyperbolic spaces. In non-hyperbolic settings, we need alternative ways to quantify the growth and comparison of functions analogous to metric potentials.
Absence of a Canonical Boundary: The Gromov boundary plays a crucial role in analyzing the asymptotic behavior of hyperbolic groups. Non-hyperbolic spaces may lack a canonical boundary notion, making it difficult to study asymptotic properties.
Potential Approaches and Considerations:

Restricting the Class of Spaces: One approach is to focus on specific classes of non-hyperbolic spaces that share some structural similarities with hyperbolic spaces. Examples include:

Relatively Hyperbolic Groups: These groups exhibit hyperbolic-like behavior outside certain well-defined subgroups. Adapting the joint translation spectrum to this setting might involve understanding its behavior with respect to these subgroups.
CAT(0) Spaces: These spaces have non-positive curvature, which provides some geometric control. However, defining appropriate analogs of metric potentials and understanding their asymptotic behavior would be crucial.

Alternative Notions of Growth: Instead of relying solely on translation lengths, we could explore alternative ways to quantify the growth of group actions on non-hyperbolic spaces. This might involve considering growth rates of volumes, diameters of orbits, or other geometric invariants.
Coarse Geometric Techniques: Coarse geometry provides tools to study large-scale properties of spaces. Adapting these techniques might help define a meaningful notion of the joint translation spectrum in a coarse sense.
Key Considerations:

Meaningful Interpretation: Any extension of the joint translation spectrum to non-hyperbolic spaces should have a clear and meaningful interpretation in terms of the geometry and dynamics of the group action.
Robustness: The definition should be robust under quasi-isometries or other appropriate equivalence relations between spaces.
Computational Aspects: Investigating the computational complexity of computing or approximating the joint translation spectrum in non-hyperbolic settings would be essential.

Could there be alternative geometric constructions besides Manhattan manifolds that provide insights into the joint translation spectrum?

Yes, there could be alternative geometric constructions that shed light on the joint translation spectrum. Here are a few possibilities:

Generalized Convex Bodies: The joint translation spectrum is a convex body. Exploring other types of convex bodies or polytopes associated with group actions could reveal further connections. For instance:

Moment Polytopes: These polytopes arise in geometric invariant theory and symplectic geometry. Investigating whether moment polytopes can be associated with group actions and how they relate to the joint translation spectrum could be fruitful.
Newton Polytopes: These polytopes are defined for polynomials and can encode information about their roots. Exploring if analogs of Newton polytopes can be constructed for functions related to metric potentials might provide new insights.

Geometric Realizations in Symmetric Spaces: Given a suitable representation of a group into a higher-rank symmetric space, one could try to realize the joint translation spectrum as a geometric object within that space. This might involve considering projections of flats or other geometric structures.
Combinatorial Constructions: For groups acting on geometric objects with combinatorial structures, such as buildings or complexes, one could explore combinatorial constructions that capture the asymptotic information encoded by the joint translation spectrum. This might involve analyzing galleries, apartments, or other combinatorial substructures.
Key Considerations for Alternative Constructions:

Duality: The Manhattan manifold exhibits a duality relationship with the joint translation spectrum. It would be interesting to see if alternative constructions also possess such duality properties.
Geometric and Dynamical Interpretation: The chosen construction should have a clear geometric or dynamical interpretation in the context of the group action.
Computational Tractability: Ideally, the alternative construction should be computationally tractable, allowing for the development of algorithms to compute or approximate the joint translation spectrum.

What are the implications of this research for understanding the computational complexity of problems related to group actions and metric spaces?

This research on the joint translation spectrum and Manhattan manifolds has several implications for understanding the computational complexity of problems related to group actions and metric spaces:

New Hard Problems: The results suggest that computing the joint translation spectrum, even for restricted cases like word metrics, is likely computationally hard. This is because:

Polytope Structure: The joint translation spectrum for word metrics is a polytope, and determining the facets of polytopes is known to be computationally challenging in general.
Connection to Ergodic Optimization: The relationship between the joint translation spectrum and ergodic optimization problems, which are often computationally hard, further supports this view.

Approximation Algorithms: While exact computation might be difficult, the geometric insights from Manhattan manifolds and the connection to random walks could lead to the development of efficient approximation algorithms for the joint translation spectrum. These algorithms could be particularly useful in applications where precise values are not essential.

Complexity Classes: This research could contribute to a more refined understanding of complexity classes related to geometric group theory. For example, it might help classify the complexity of problems like:

Determining Membership: Deciding whether a given vector belongs to the joint translation spectrum.
Approximating the Spectrum: Finding an approximation of the joint translation spectrum within a given error bound.

Connections to Other Fields: The tools and techniques developed in this research, such as the use of geodesic currents and the analysis of Manhattan manifolds, could have implications for computational problems in related fields like:

Dynamical Systems: Understanding the complexity of invariant measures and other dynamical invariants.
Geometric Topology: Analyzing the complexity of geometric structures on manifolds and their deformations.
Overall, this research highlights the intricate interplay between geometry, dynamics, and computation in the study of group actions. It suggests new avenues for investigating the computational complexity of fundamental problems in geometric group theory and related areas.